Factor completely: 2s² + 9s + 7.
Understand the Problem
The question is asking to factor the quadratic expression given, specifically 2s² + 9s + 7, completely.
Answer
The completely factored form is $(2s + 7)(s + 1)$.
Answer for screen readers
The completely factored form of the expression $2s^2 + 9s + 7$ is: $$(2s + 7)(s + 1)$$
Steps to Solve
- Identify the quadratic expression
The expression we need to factor is $2s^2 + 9s + 7$.
- Multiply the coefficient of $s^2$ by the constant term
Multiply the coefficient of $s^2$ (which is 2) by the constant term (which is 7): $$ 2 \times 7 = 14 $$
- Find two numbers that multiply to 14 and add to 9
We need to find two numbers that multiply to 14 and add to 9. The numbers that work are 7 and 2, since: $$ 7 \times 2 = 14 \quad \text{and} \quad 7 + 2 = 9 $$
- Rewrite the middle term
Rewrite the quadratic expression by breaking the middle term (9s) using the two numbers found: $$ 2s^2 + 7s + 2s + 7 $$
- Factor by grouping
Next, group the terms: $$ (2s^2 + 7s) + (2s + 7) $$
Factor out the common factors in each group: $$ s(2s + 7) + 1(2s + 7) $$
- Factor out the common binomial factor
Now, factor out the common binomial factor $(2s + 7)$: $$ (2s + 7)(s + 1) $$
The completely factored form of the expression $2s^2 + 9s + 7$ is: $$(2s + 7)(s + 1)$$
More Information
Factoring quadratics is a useful skill in algebra. It allows us to find roots of equations and simplify expressions. The technique used here involves recognizing the need to break down the middle coefficient into two numbers that fit the requirements of multiplication and addition.
Tips
- Incorrectly identifying the numbers: Make sure the two numbers multiply to the product of the leading coefficient and constant, and also add to the middle coefficient.
- Forgetting to check the signs of the values: Always validate that the signs of the factors align with the given expression.
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